How Do You Determine the Probability of Two Independent Events Happening?

The probability involved in rolling one die is pretty simple. Add more dice, or roll it more than once, and things get a bit more complicated.

Many games involve chance. Even the most skillfull games usually involve some element of luck. Sometimes this involves drawing cards from a deck, but more often than not it involves rolling a die.

The basic probability of rolling a die is very simple. You've got (in most cases) six sides. Each side has a 1 in 6 chance of coming up.

Your probability of rolling a six? 1 in 6 (16.67%). Your probability of rolling a three? 1 in 6 (16.67%). Your probability of rolling a five or six? 2 in 6 (33.33%).

What About Rolling Two Dice Together?

Yes, this is where things get slightly more complicated. Basic probability involves manipulating three numbers - the chance of success, the chance of failure, and the total number of outcomes. A chance of an event occuring is usually expressed as a fraction - the number of successes over the total number of outcomes (or, for the chance of failure, the number of failures over the total number of outcomes).

When you're rolling one die, the total number of outcomes is the number of sides on a die. If you have a six sided die, there are six total outcomes. If you define a success as rolling a six, then your number of successes is "one" - you only have one chance to get a six on the entire die. That gives you a probability of 1/6 (16.67%).

The same principle applies to rolling two dice - you need to determine the number of possible successes and the total number of outcomes.

If you roll two six-sided dice together, you have thirty-six possible outcomes. Go ahead and list them all if you want. Or, use a simple mathematical trick - multiply the possible outcomes on the first die by the possible outcomes on the second die (6 x 6 = 36). If you had two eight sided dice, there would be 64 possible outcomes.

Let's say a success occurs if you roll a total of a twelve. How many chances are there? One! You'd have to roll a six on both dice. The overall probability of rolling a twelve with two six sided dice is 1/36 (2.8%).

You can also look at this another way. You have two independent events occurring, and you want to know the probability of both events happening at the same time. In other words, you're rolling two dice. Neither one has an influence on the other. You want to know the probability that both of the dice will do something (i.e. come up six, or be four or higher).

The mathematical way to figure this out is to take the probability of the first event (rolling a six, 1/6) and multiply it by the probability of the second event (rolling a six, 1/6). 1/6 * 1/6 = 1/36. If you wanted to know the probability that each die would be four or higher, you'd have 3/6 * 3/6 = 9/36 (25%).

What If I Only Need One Die to Be a Six?

There are also a lot of times in games where you have more than one chance to do something. Let's say you need to roll a five or better to succeed. Rolling one die, that's a probability of 2/6 (1/3 or 33.33%). But what if you have two chances to do it? It doesn't matter if both dice come up as five or better, you just care if one or the other does.

There's a simple mathematical formula for figuring this out. Add the chance of the first event succeeding (2/6) to the chance of the second event succeeding (2/6). Then, subtract the chance that they both happened at the same time (2/6 * 2/6 = 4/36). Once you simplify all of the fractions (2/6 + 2/6 - 4/36) you're left with an overall probability of 20/36 or 55.56%. While you only had a roughly 33% chance of succeeding with one dice roll, you've got better than a fifty percent chance of succeeding with two.

To take one more example, let's say you have two chances to roll a 4 or better. There's a 50 - 50 chance that you'll roll a 4, 5, or 6. Add the probabilities together (1/2 + 1/2) and subtract the chance that they both happen simulataneously (1/2 * 1/2 = 1/4). You're overall probability is (1/2 + 1/2 - 1/4 = 3/4 or 75%).

Mathematical Recap

While the examples are certainly not exhaustive, this should be enough for you to figure out many of the basic probabilities involved in rolling dice. Just remember the two basic formulas:

The Probability that Both Dice Do Something = Number of Successes / Number of Outcomes

The Probability that Either Die Does Something = (Probability that the First Succeeds) + (Probability that the Seconds Succeeds) - (Probability that they Both Succeed)

You should also remember this basic principle of probability. No matter how many chances you have (i.e. rolling a die 2, 3, 4, or more times), you never have a 100% chance of success! Notice in the last example (rolling a die twice trying to get a 4+), you only have a 75% chance of failure! This often seems counterintuitive, and if you don't work out the probability of the event you're likely to over-estimate how often you will actually come out successful.